On An Unstable Thin-film Equation With Convective Term

We studied the initial boundary value problem of the unstable thin-film equation (?) where ?(?)RN is an arbitrary bounded domain(N ? 1)with boundary(?)? of the class C1.1,QT =(0,T)×?,n>0,m?R,a0>0,and a1 ? R.Because of the degeneracy,we first considered the regularized problem ht + div(f?(h))(a0??h +a1D?"(h)?h)=(?)·?B(h),? QT,h = ?h = 0,?(?)?(0,T),h(x,0)= h0,??(x),where (?)Using the Faedo-Galerkin method,we proved the existence of solutions for the regularized parabolic problem.By means of energy and entropy estimates,using the Poincare inequality and the Young inequality,we proved the existence of non-negative weak solutions and strong solutions for the initial boundary value problem.